The number of dots given are 1, 4, 9, 11, 16, 25, ...
The no. of dots are in the pattern 1 \(\times\) 1, 2 \(\times\) 2, 3 \(\times\) 3, 4 \(\times\) 4, 5 \(\times\) 5 .... 10 \(\times\) 10
\(\therefore \) The number of dots in the 10^th figure = 100.

The single portions  A, B, C, D, E, F, G and H are 8 triangles. 
Taking the combination of 2 E & F, F & G, & H and E & H are 4 triangles.
There are 8 + 4 = 12 triangles.

6 possibilities,
i) R - B - R - B  
ii) R - R - B - B 
iii) B - R - R - B 
iv) B - R - B - R  
v) B - B - R - R  
vi) R - B - B - R
(R- RED, B- BLUE)

There are 12 straight lines used which is minimum.

The given star can be viewed as two magical triangular as.

Now the required arrangement is

Some other arrangements are

Here there are even number of terms.
Also we know that 1 + 6 = 7, 2 + 5 = 7, 3 + 4 = 7;
so placing 7 at the centre, and the pairs (1, 6) (2, 5) and (3, 4) at the opposite ends we get, the answer.

The odd numbers between 1 to 17 are 1, 3, 5, 7, 9, 11, 13, 15, 17.
Step 1: Place the smaller numbers 1, 3 : 5 on the comers.

Step 2: Arrange another set of smaller numbers 7, 9 and 11 on each side.

Step 3: Arrange the remaining numbers 13, 15, 17 to give the total 30.

Magic sum = 30.

 

 

 

 

 

(i) Yes
(ii) 5
(iii) 
     
Sums are 17, 19, 20, 21 and 23.

The number of dots given are 1, 3, 6, 10, 15, ...

Pattern	Number of dots
1	1
2	3
3	6
4	10
5	15

1^st number = 1
2^nd number = 1 + 2 = 3
3^rd number = 3 + 3 = 6
4^th number = 6 + 4 = 10
5^th number = 10 + 5 = 15
6^th number will be 15 + 6 = 21
7^th number will be 21 + 7 = 28
8^th number will be 28 + 8 = 36
9^th number will be 36 + 9 = 45
10^th number will be 45 + 10 = 55
Number of dots in the 10^th figure = 55.

(iii) Every part A, B, C, D, E, F, G, H, I, J, K, L, M, N, O and P are 16 triangles.
Combining two at a time. A & B, B & C, C & D, D & A, J & K, L & M, N & O, P & I are 8 triangles.
Taking the combination of 4, we have I, J, E & A; K, L, F and B, M, N, G and C, P, O, H and D are 4 triangles.
Taking 8 parts together we have QRS, RST, STQ, and TRQ are 4 triangles.
\(\therefore \) Total triangles = 16 + 8 + 4 + 4 = 32 triangles.

The smaller portions A, B, C, D, E, F, G and H are 8 triangles.
Combining 2 at a time B & C, D & F, F & G, A & H are 4 triangles.
Taking 4 at a time Combination of B, C, D & E;
D, E, F and G;
F, G, H and A
H, A, Band Care 4 triangles.
Total triangles = 8 + 4 + 4 = 16.